How to create a "Rounded Grids" effect like in MagicaVoxel
UPDATED ANSWER 1

Yeah, it might be a bug, you could report it into the bug tracker.

Here's the best workaround I could find:


set the influence to diffuse
create a second uv map (a quick Lightmap Pack should be fine)
bake into a uv-mapped texture (Bake mode: Textures)
now you can use this texture (in Cycles as well!) for the bumps


Cycles renders look better anyway, so you could consider this an improvement instead of a sad workaround :)

ANOTHER, EVEN BETTER, WAY TO DO IT

An hour later an even faster method occurred to me, where you don't need Blender Internal at all:


export-import as ply (the smaller faces in ply correspond to the original "grid")
add a second uv map. 
when you actually unwrap it, just select "Reset" - this assigns the full UV space to each face
use your original bump texture, which will be assigned to each grid face


ORIGINAL ANSWER 

The simplest way to do this is to add the Bevel Modifier to all of your cubes - you have a lot of control without messing with materials and textures. Increase the segments of the beveling until you are happy with the result.

The following question can help you to do this efficiently: Add the Same Modifier to Multiple Objects at Once?

If the OBJ format export-import creates triangles from the quads, then the bevel modifier could have problems. See this for converting back to quads: How do I convert a triangle mesh to a quads one?